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REVIEW 3 major objections 3 minor 26 references

On inequalities for A-numerical radius of operators

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Pith's one-line read For a strictly positive weight A, the A-numerical radius squared is bounded below by one quarter and above by one half of the symmetrized A-product norm.

desk verdict Main A-numerical radius inequalities hold and genuinely improve on Zamani; Theorem 2.16 has a fixable gap, and the paper leans on unpublished lemmas, but it deserves a referee. read the letter →

arxiv 1908.11182 v2 pith:JKQMI4TL submitted 2019-08-29 math.FA

Pintu Bhunia , Kallol Paul , Raj Kumar Nayak This is my paper · ORCID
classification math.FA MSC 47A1247A3047A63
keywords A-numericalradiussemi-HilbertianspacepositiveoperatorA-adjointmatrixnumericalinequalityA-Crawfordnumberpowermean
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper works in the semi-Hilbertian setting where a positive operator A induces its own inner product and its own numerical radius w_A(T). Its central claim is that for strictly positive A, the square of w_A(T) is sandwiched between one quarter and one half of the A-norm of the symmetrized product T $T^{{♯_A}}$ + $T^{{♯_A}}$ T, where $T^{{♯_A}}$ is the A-adjoint of T. The lower bound is new in this weighted setting, and the whole two-sided estimate reduces to the classical Hilbert-space numerical radius inequality when A is the identity. The paper also proves fourth-power refinements, exact values for operators squaring or cubing to zero, and bounds for products of operators and for 2×2 block matrices with A repeated on the diagonal. A sympathetic reader should care because these inequalities locate the A-numerical radius, a quantity that controls the A-operator seminorm and appears in approximation and spectral estimates for operators on weighted spaces.

What carries the argument

The central mechanism is the decomposition of an operator into its A-real and A-imaginary parts, H_θ=Re_A($e^{{iθ}}$T)=($e^{{iθ}}$T+$e^{{-iθ}}$$T^{{♯_A}}$)/2 and K_θ=Im_A($e^{{iθ}}$T), together with the identity $H_θ^{2}$+$K_θ^{2}$=\frac12\operatorname{diag}(X $X^{{♯_A}}$+$Y^{{♯_A}}$Y,\,$X^{{♯_A}}$X+Y $Y^{{♯_A}}$) for the block matrix [[O,X],[Y,O]]. Lemma 2.3 reduces w_A(T) to sup_θ \|H_θ\|_A, and Lemma 2.5 lets the authors compare A-norms of A-positive operators ordered by A-positivity. These pieces turn the numerical radius into a norm computation for a symmetrized product, which is what the two-sided bounds express.

What would settle it

Let A=diag(1,2,3) on $C^{3}$, pick a specific T such as a 3×3 matrix with a single nonzero superdiagonal entry, form P=T $T^{{♯_A}}$+$T^{{♯_A}}$ T, and numerically compare $w_A^{2}$(T) with (1/4)||P||_A; if the inequality fails for any such T, the main claim is false, and the same example can be used to check the power-mean step with r=2 and r=3.

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Extended reading notes

Core claim

On its own terms, the paper establishes that for A>0 and T in B_A(H), every such operator satisfies $$ \frac14 \|T $T^{{\sharp_A}}$ + $T^{{\sharp_A}}$ T\|_A \le $w_A^{2}$(T) \le \frac12 \|T $T^{{\sharp_A}}$ + $T^{{\sharp_A}}$ T\|_A. $$ The upper bound matches the earlier A-numerical radius inequality, while the lower bound is new and becomes the classical Hilbert-space inequality when A=I. The same mechanism, applied to the block matrix [[O,X],[Y,O]] with B=diag(A,A), gives two-sided B-numerical radius inequalities controlled by the A-norms of X $X^{{♯_A}}$+$Y^{{♯_A}}$Y and $X^{{♯_A}}$X+Y $Y^{{♯_A}}$. The paper further claims that fourth-power identities refine these bounds, that operators with $T^{2}$=0 or $T^{3}$=0 admit exact A-numerical radius formulas, and that product-operator inequalities improve on earlier bounds by subtracting nonnegative A-Crawford terms.

Load-bearing premise

The r-th power estimate in Theorem 2.16 assumes that averaging two A-positive operators and then taking the A-norm behaves like the scalar power mean, namely ||(S+U)/2||_A^r ≤ ||((S^r+U^r)/2)||_A for every r≥1, but the paper justifies this only by saying the scalar functions t^r and $t^{{1/r}}$ are convex and concave.

Editorial extensions

If this is right

  • For strictly positive A, w_A^2(T) has a concrete lower bound, so w_A(T) cannot drop below half the square root of the symmetrized A-product norm; this closes a gap left by earlier upper-only estimates.
  • Setting A=I recovers the classical Hilbert-space inequality, so the result is a genuine generalization rather than an isolated weighted-space estimate.
  • The fourth-power and r-th-power bounds give progressively sharper control of w_A(T), including exact values when T^2=0 or T^3=0.
  • The block-matrix inequalities apply to B=diag(A,A), so any 2×2 operator matrix with A-bounded entries has its B-numerical radius controlled by the A-norms of the two diagonal combinations X X^{♯_A}+Y^{♯_A}Y and X^{♯_A}X+Y Y^{♯_A}.
  • The product inequalities of Section 3 strengthen the earlier bounds for w_A(XY) by subtracting nonnegative A-Crawford terms, so the estimates become strictly better except in degenerate cases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the same two-sided picture should survive for A that is strictly positive only on a dense subspace, because the proof relies on order and monotonicity rather than on invertibility of A; testing whether the lower bound persists when the range of A is not closed would be a natural extension.
  • The 2×2 block method suggests an iteration: applying the same identity to block matrices with more entries could yield A-numerical radius bounds for n×n operator matrices, with the diagonal combinations replaced by sums over each row.
  • The exact equality when T^2=0 invites a characterization question: whether w_A(T)=1/2√\|T T^{♯_A}+T^{♯_A} T\|_A characterizes two-step nilpotents in the A-setting, or whether the paper's own counterexample for the converse is typical.
  • The power-mean step in Theorem 2.16 is the only place the argument exceeds scalar convexity; if a full proof is supplied, the r-th power bound would immediately strengthen all the fourth-power corollaries to arbitrary powers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies A-numerical radius inequalities for operators on a Hilbert space equipped with a positive-semidefinite inner product induced by a positive operator A. The main results are two-sided bounds for the A-numerical radius of 2x2 operator matrices, a new lower bound in Corollary 2.7 that recovers Kittaneh's Hilbert-space inequality when A=I, an upper bound for w_A^4(T) and w_A^3(T), a power inequality w_A^{2r}(T) in Theorem 2.16, lower bounds involving the A-Crawford number, and product inequalities in Section 3. The paper explicitly shows that several bounds reduce to or improve on results of Zamani and Kittaneh.

Significance. If the proofs are completed, the results give useful refinements of the semi-Hilbertian numerical-radius literature: Corollary 2.7 is a genuinely new lower bound in the A-setting, and the product inequalities in Corollary 3.7 improve on Zamani's bounds. The paper has the virtue of deriving explicit, checkable estimates and of demonstrating the reduction to the known A=I cases. The significance is incremental rather than revolutionary, but the results are natural and would be of interest to researchers working on numerical radius inequalities and semi-Hilbertian operator theory.

major comments (3)
  1. [Theorem 2.16, page 9] The proof uses the step ||(T^♯AT+TT^♯A)/2||_A^r ≤ ||((T^♯AT)^r+(TT^♯A)^r)/2||_A, justified only by saying that t^r is convex. Scalar convexity alone does not justify this A-operator power-mean inequality; it requires an A-Jensen type result, e.g., ⟨S^r x,x⟩_A ≥ ⟨Sx,x⟩_A^r for A-positive S and r≥1, applied to the A-inner-product completion. Without a proof or reference for this inequality, the derivation of Theorem 2.16 is incomplete. This is repairable by adding a short lemma, but as written it is a genuine gap. Please also state explicitly whether the statement requires A>0 or any range-closedness condition for the A-power-mean inequality to hold.
  2. [Lemma 2.4 and Theorem 3.1] The block-adjoint formula and the B-unitary invariance w_B(U^♯B T U)=w_B(T) are quoted from the unpublished preprint [10, Lemma 3.1 and Lemma 3.8]. These facts are load-bearing for Lemma 2.4(ii)-(iv) and, through Lemma 2.4, for Theorem 3.1 and related product bounds. Since [10] is not a published source, the paper should either prove these facts in a short lemma or replace the citation with a published reference. This is a completeness issue that affects several results.
  3. [Theorems 2.6, 2.14, 2.18] The proofs implicitly use the identity ||H^n||_A = ||H||_A^n for A-selfadjoint H (for n=2 in Theorem 2.6 and n=3,4 in Theorems 2.14 and 2.18). This is true because H^2 is A-positive and ||H^2||_A = sup_x ||Hx||_A^2 = ||H||_A^2, and similarly for higher powers, but the identity is never stated. Adding a one-sentence justification would make these arguments fully self-contained.
minor comments (3)
  1. [Lemma 2.4(iv)] The verification that U is B-unitary and the computation U^♯B T U = diag(X-Y,X+Y) are dismissed as 'an easy calculation'; a few lines of details would help the reader.
  2. [Abstract and formatting] There are minor typographical issues, including the stray space in 'existing ones ,' in the abstract, the use of '0' instead of 'O' in the proof of Theorem 2.9, and 'converse is not true, that is' in Remark 2.15, which should read 'i.e.'.
  3. [Remark 2.13 and Remark 2.19] The numerical comparisons in these remarks would be easier to follow if the constants were derived in one or two lines rather than asserted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inequalities are derived from definitions via elementary estimates, and the self-cited technical lemmas are standard, parameter-free tools that do not embed the target results.

full rationale

The central claims (Theorem 2.6, Corollary 2.7, Theorem 2.9, Theorem 2.12, Theorem 2.14, Theorem 2.16, Theorem 2.18, and the product inequalities in Section 3) are derived from the definitions of the A-numerical radius, A-adjoint, A-seminorm, and elementary algebraic manipulations. There are no fitted parameters, no data-driven constants, and no quantity called a prediction that is in fact an input. Corollary 2.7 reduces to Kittaneh's Hilbert-space inequality when A=I, which confirms that the result carries independent content rather than being an identity. The self-citations are [10, Lemma 3.1] and [10, Lemma 3.8], used for the block form of the B-adjoint and B-unitary invariance of the B-numerical radius; these are standard, parameter-free facts whose assumptions do not include any of the paper's target inequalities, so they do not make the derivation circular. Remarks 2.11(i), 2.13, and 3.5 cite the authors' own work only for comparison or for already-published auxiliary results, not as the load-bearing premise of a central theorem. Remark 2.13 uses the paper's own Corollary 3.3 to establish an improvement, but this is a legitimate internal comparison, not a hidden assumption of the derived inequality. The only serious gap is in Theorem 2.16, where the proof replaces ||(T^sharp_A T + T T^sharp_A)/2||_A^r by ||((T^sharp_A T)^r + (T T^sharp_A)^r)/2||_A with only a remark on convexity of t^r; that step needs a cited or proved A-operator power-mean inequality. This is a missing justification and a possible correctness risk, not a circular step: the inequality is not identical to the theorem's conclusion, and the theorem is not invoked to prove the step. No uniqueness theorem is imported from the authors' prior work, no ansatz is smuggled in via citation, and no known empirical pattern is merely renamed. Accordingly, the paper is not circular; the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper assumes standard functional analysis background, including C*-algebras, Douglas range inclusion, and Moore-Penrose inverses, and strictly positive A for most theorems. It also relies without proof on two lemmas from the authors' own preprint [10] and on a nontrivial operator power mean inequality in Theorem 2.16 that is justified only by a scalar convexity remark.

assumptions (3)
  • domain assumption The B-unitary invariance identity w_B(U^sharp T U) = w_B(T) from [10, Lemma 3.8] and the block A-adjoint formula T^sharp B = (T_ji^sharp A) from [10, Lemma 3.1] hold.
    Used in Lemma 2.4 and Theorem 3.1; these results are cited to the authors' own arXiv preprint 2001.04522 and are not proved in this paper.
  • standard math For positive A-operators S = T^sharp A T and U = T T^sharp A, the power mean inequality ||(S+U)/2||_A^r <= ||(S^r+U^r)/2||_A holds for all r >= 1.
    Invoked in Theorem 2.16 to pass from scalar convexity to an operator norm inequality; the paper gives no proof or citation for this matrix result.
  • domain assumption Strict positivity A > 0 is assumed in Corollaries 2.7 and 2.10 and throughout Section 3.
    This makes A-adjoints unique, makes the A-seminorm a norm, and lets the authors use P_A = I; the results are not claimed for singular A in those statements.

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Pith. "Pith review of On inequalities for A-numerical radius of operators." pith.science (2026). https://pith.science/paper/JKQMI4TL

@misc{pith2026190811182,
  author       = {Pith},
  title        = {Pith review of: On inequalities for A-numerical radius of operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JKQMI4TL}},
  note         = {Machine review of arXiv:1908.11182}
}
abstract

Let $A$ be a positive operator on a complex Hilbert space $\mathcal{H}.$ We present inequalities concerning upper and lower bounds for $A$-numerical radius of operators, which improve on and generalize the existing ones, studied recently in [A. Zamani, A-Numerical radius inequalities for semi-Hilbertian space operators, Linear Algebra Appl. 578 (2019) 159-183]. We also obtain some inequalities for $B$-numerical radius of $2\times 2$ operator matrices where $B$ is the $2\times 2$ diagonal operator matrix whose diagonal entries are $A$. Further we obtain upper bounds for $A$-numerical radius for product of operators which improve on the existing bounds.

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